The train marshalling problem

The problem considered in this paper arose in connection with the rearrangement of railroad cars in China. In terms of sequences the problem reads as follows: contains all the elements of S p(1) , then all elements of S p(2) , : : :, etc., and nally all elements of S p(t). The aim of this paper is to show that the decision problem: \Given numbers n; k and a partition S of f1; 2; :::; ng, is K (S) k?" is NP-complete. In light of this, we give a general upper bound for K (S) in terms of n.